The Philosophy of Mathematics Today

Fortunately Tarski has shown when a relation or property is logical, namely
when it is invariant under all permutations of any domain of individuals
over which it is defined.13 Transitivity is a logical property of relations for
example, and it applies to many relations that are not themselves logical,
such as the relation acquired later than defined over the books in my library.
That an ordering relation on a linearly ordered set--whatever the relation
is in concreto--is an ordering may be an empirical fact, but to be a linear
ordering is to have that logical property, so the order type of a linearly
ordered set is a structure, where only the cardinality and the logical properties of being a linear ordering are invariant. Consider the pordinal 3v.
What makes it a pure structure is that it is a partial ordering on three elements such that there is one element preceding the other two, neither of
which precedes the other, and these two obviously have no successors.
These are all logical properties.

So we have an idea what it means to be a pure simple structure. Now consider a complex structure $, whose immediate elements are either points or
themselves pure structures. Then if the structure obtained from $ by replacing each element by a point is itself a pure simple structure then $ is itself
pure. The finite structure of structures (suppressing the dots) in Fig. 19.3,
showing the inclusion of pordinals in others, is a pure structure because each
pordinal is a pure simple structure and their inclusion structure is itself a
pordinal somewhat higher up the pordinal hierarchy, and likewise pure.
There is a question mark over how this iteration, determining what purity
is, may be continued into the transfinite.

6. SUMMARY

I indicated how Frege-abstraction delivers lowish-level abstracta, not just
mathematical ones, and that mathematical structures may be obtained by
judicious choice of equivalence relation. The idea that mathematics is the
theory of pure structures possibly only works for pure mathematics, but an
attempt was made to elucidate purity of structure. I suggested how the
adjustive conception of abstraction, and being given leeway to concatenate
and combine structures, suggest a discriminating and powerful ontology of
structures, and noted several imponderables as to how far structures may
be generated by iterative combination: well-known paradoxes indicate
there are limits to the procedure. Pure abstraction on its own appears too
cardinally impoverished to deliver the structures mathematicians find interesting. One of the important issues in structuralism which must be faced
concerns the identity conditions for structures and points, of which Resnik

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